Finite Group Databasebeta

← #27276485

Subgroups — #27276485

Order 13568.

692 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275536C211yesmaximal, characteristic, complemented
2#20273441C211yesmaximal, characteristic, complemented
3#20273335C212yesmaximal, complemented
4#20273253C212yesmaximal, complemented
5C53×#3544C211yesmaximal, characteristic
6#16810680C2212yes—
7C2×#13133141—22no—
8C2×#13133128C2211yescharacteristic
9#16810489C2211yescharacteristic
10#16810372—22no—
11#16810365—22no—
12#16810363—22no—
13#16810360—22no—
14#16810357—22no—
15C53×#563—22no—
16Q82×C53C411yescharacteristic, complemented
17C53×#555C411yescharacteristic, complemented
18C53×#554C412yescomplemented
19C53×#552—22no—
20C53×#547—22no—
21C53×#544—22no—
22C53×#542—22no—
23C53×#536—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C53×#529C2212yes—
26C22×#88322—21no—
27C2×C4⋊Dic53—21no—
28C4⋊#88322—22no—
29#13133141—44no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128—22no—
33#13133124—22no—
34C2×#41×C53—22no—
35C22×Q8×C53D412yes—
36C4⋊Q8×C53—22no—
37C4⋊Q8×C53—22no—
38C4⋊Q8×C53—22no—
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53C2×C411yescharacteristic
41C4⋊Q8×C53C2×C411yescharacteristic
42C4⋊Q8×C53D412yes—
43#126×C53—24no—
44#126×C53—22no—
45#125×C53—24no—
46#125×C53—22no—
47#125×C53D412yes—
48#124×C53—22no—
49#124×C53—22no—
50#124×C53D412yes—
51#123×C53—24no—
52#123×C53—24no—
53#123×C53—22no—
54C22⋊Q8×C53—24no—
55C22⋊Q8×C53—24no—
56C22⋊Q8×C53—24no—
57C22⋊Q8×C53—22no—
58C22⋊Q8×C53D412yes—
59#121×C53—22no—
60#121×C53D412yes—
61C4×Q8×C53—22no—
62C4×Q8×C53C2×C412yes—
63C4×D4×C53—22no—
64C4×D4×C53—22no—
65C4×D4×C53C2×C412yes—
66#117×C53—22no—
67#117×C53—22no—
68#117×C53—22no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C42×C53C2311yescharacteristic
72#41×C53—82no—
73#41×C53—82no—
74C2×Q8×C53—22no—
75C2×Q8×C53—22no—
76C2×Q8×C53—42no—
77C2×Q8×C53—42no—
78C2×Q8×C53C22⋊C412yes—
79C2×Q8×C53C22⋊C412yes—
80C2×Q8×C53C22⋊C412yes—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53C22⋊C412yes—
84C22×C4×C53—22no—
85C22×C4×C53—24no—
86C22×C4×C53—22no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D412yes—
89C22×Dic53—41no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
92C4⋊C4×C53—24no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53C22⋊C412yes—
101C4⋊C4×C53C22⋊C412yes—
102C22⋊C4×C53—24no—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—22no—
105C22⋊C4×C53—22no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C42×C53—22no—
110C42×C53—22no—
111C42×C53—22no—
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D412yes—
114C42×C53C2×D411yescharacteristic
115C4⋊Dic53—42no—
116C2×#88322—22no—
117C2×#88322—41no—
118C2×#88322—41no—
119C2×#88322—42no—
120C23×C53—22no—
121C23×C53#12011yescharacteristic
122Q8×C53—42no—
123Q8×C53—42no—
124Q8×C53—42no—
125Q8×C53—82no—
126Q8×C53—82no—
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—42no—
130D4×C53—82no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—42no—
139C2×C4×C53—44no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—84no—
143C2×C4×C53C2×C22⋊C412yes—
144C2×C4×C53C2×C22⋊C411yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12012yes—
147C2×Dic53—82no—
148C2×Dic53—81no—
149#88322—42no—
150#88322—82no—
151#8498—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53—42no—
156C22×C53#37911yescharacteristic
157C22×C53#45311yescharacteristic
158C22×C53#45511yescharacteristic
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—42no—
164C4×C53—42no—
165C4×C53—42no—
166C4×C53—42no—
167C4×C53—42no—
168Dic53—161no—
169#3544D5311yescharacteristic
170#1960—531no—
171#1558—531no—
172#1532—532no—
173#1512—532no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—82no—
177C2×C53#179211yescharacteristic
178C2×C53#179612yes—
179#563—22no—
180Q82Dic5311yescharacteristic, complemented
181#555Dic5311yescharacteristic, complemented
182#554Dic5312yescomplemented
183#552—22no—
184#547—22no—
185#544—22no—
186#542—22no—
187#536—22no—
188C2×C4⋊Q8C2×D5311yescharacteristic
189#529C2×D5312yes—
190#432—532no—
191C2×#103—1062no—
192C2×#97—531no—
193#382—531no—
194#333—1062no—
195#330—1062no—
196#329—1062no—
197#328—1062no—
198#327—1062no—
199C53#849811yescharacteristic, complemented, Sylow
200C2×#41—22no—
201C22×Q8C22⋊D5312yes—
202C22×C8—1061no—
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8C2×Dic5311yescharacteristic
208C4⋊Q8C2×Dic5311yescharacteristic
209C4⋊Q8C22⋊D5312yes—
210#126—24no—
211#126—22no—
212#125—24no—
213#125—22no—
214#125C22⋊D5312yes—
215#124—22no—
216#124—22no—
217#124C22⋊D5312yes—
218#123—24no—
219#123—24no—
220#123—22no—
221C22⋊Q8—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8C22⋊D5312yes—
226#121—22no—
227#121C22⋊D5312yes—
228C4×Q8—22no—
229C4×Q8C2×Dic5312yes—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4C2×Dic5312yes—
233#117—22no—
234#117—22no—
235#117—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—1061no—
239C2×C42C22×D5311yescharacteristic
240C4⋊C8—1062no—
241#103—2124no—
242C22⋊C8—1062no—
243#97—1062no—
244#97—1062no—
245#95—1062no—
246#41—82no—
247#41—82no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8C22⋊Dic5312yes—
253C2×Q8C22⋊Dic5312yes—
254C2×Q8C22⋊Dic5312yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C22⋊Dic5312yes—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—2121no—
262C22×C4C2×C22⋊D5311yescharacteristic
263C22×C4C2×C22⋊D5312yes—
264C2×C8—1062no—
265C2×C8—2122no—
266C2×C8—2121no—
267C2×C8—2121no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—2122no—
279C4⋊C4C22⋊Dic5312yes—
280C4⋊C4C22⋊Dic5312yes—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—22no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42C22×Dic5311yescharacteristic
292C42C2×C22⋊D5312yes—
293C42C2×C22⋊D5311yescharacteristic
294C23—22no—
295C23#1313318611yescharacteristic
296Q8—42no—
297Q8—42no—
298Q8—42no—
299Q8—82no—
300Q8—82no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—4242no—
318C2×C4—4241no—
319C2×C4C2×C22⋊Dic5312yes—
320C2×C4C2×C22⋊Dic5311yescharacteristic
321C2×C4#1313318612yes—
322C2×C4#1313318611yescharacteristic
323C8—4242no—
324C8—2122no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22#1681047911yescharacteristic
330C22#1681079111yescharacteristic
331C22#1681080111yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—21no—
336C4—42no—
337C4—42no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—8481no—
342C2—21no—
343C2—21no—
344C2—82no—
345C2#2027466911yescharacteristic
346C2#2027469012yes—
subgroupG/H
centerC22#16810479
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3544C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3544D53
hypocenterC53#8498
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3544

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20275536C2semidirect
2. #20273441C2semidirect
3. #20273335C2semidirect
4. #20273253C2semidirect
16. Q82×C53C4semidirect
17. C53×#555C4semidirect
18. C53×#554C4semidirect
180. Q82Dic53semidirect
181. #555Dic53semidirect
182. #554Dic53semidirect
199. C53#8498semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202755362
#202734412
#202733352
#202732532
C53×#35442
#849853